Mr. HOLDITCH, ON SMALL FINITE OSCILLATIONS. 



97 



d0 

 s^a = 2 + 2 . { (.T - a) . cos - (y - /3) . sin 0) } . — 



= 2 + 2 . J (.r - a) . cos - (y - /3) . sin 0} . R~\ 

 z.,= -2. \{a: -a).sin0+ (y - ,8) . cos J . iZ"^ - 2 . {(,t; - a) . cos - (y - /3) . sin 0) . «-= . fl, , 

 .^, = -2i?-^-2. \{a!-u).cose- (y-/3).sin0| . (R' + R'- . R, - 2R-'Rl) 

 omitting the terms which vanish ; but since U = 0, (y — /3) . cos 6 + {v — n) . sin = 0, 

 and therefore {x — a) . cos 9 — (y — fi) . s\n6 = — S. 



Hence -=i -_, 



Xf3 5 R\ 



^ S-R S.(.RR,-2R\) 

 2 ~ R^ '^ R^ 



1 2 



Also F=-— = 1, and F, = F^ = ; 

 as 



L = 



Rh.i 



(20). 



-U, kf(S).(S-R) 



g [ 16 L3»2 »2 -'J 



f As^ri ^/?, i??^.(6i?-5) 3.(^-i?) ., /(5)-]i . 



I) 



/(^) 



If the force vary as the distance, rfj . log \ = 0, and the force does not affect the cor- 







rection of the expression for the time. 



2R, Si. As 



The excess of the time of ascent = - 



3 '{S- R)K\/kRf{S)' 

 SR, A«^ 



angle 



RS-R'' 3 ' 



rT^^'^'-^-T' 



If the force be constant and act in parallel lines, S is infinite, and f(S) constant, and formula 

 (21) becomes the same as (19)- 



(1). To find the time of oscillation of a particle in the centre 

 of a hollow sphere, the force varying as the ra"" power of the dis- 

 tance, and the density being = ju . r'". 



Let QOR = <p and QCP = 6 ; then the volume of a particle 

 at Q = r^dr fiinO .dd . d(p, and its force on the particle at P, 

 where CP = w and QP = p, is 



lui .r'^-'dr . sin e . dO . d(p . p" ; 



d,E 



dp 



M . r'"*'dr .sine.de. deb .p'-r + 



d.v 



Vol.. VIII. 1'art I. 



N 



